Abstract: A class of quasi-cubic-Bernstein basis functions with a single parameter is presented,which is an extension of the cubic Bernstein basis functions defined over the triangular domain.Based on the introduced functions,we propose a method to produce the quasi-cubic-B-B parametric surfaces defined over the triangular domain with a single shape parameter.The surfaces'properties are similar with the cubic B-B parametric surfaces.In particular,when the shape parameter equals 1,they degenerate to the cubic B-B parametric surfaces.By changing the shape parameter,we can get surfaces with different shapes in invariable control net.